THE SETUP
A Number Blocks grid is N×N with target sums displayed outside each row and column. You fill in every cell with a digit from 1 to 9. Two constraints govern every placement:
- Row sums: the digits in each row must add up to the row's target.
- Column sums: the digits in each column must add up to the column's target.
Unlike Kakuro, digits can repeat within a row or column — there is no unique-digit rule. A row of three with target 12 could be 4+4+4 or 3+5+4 or any other combination summing to 12 with digits 1–9. What constrains the puzzle is the overlap between rows and columns: each cell must satisfy one row constraint and one column constraint simultaneously.
ONE-EMPTY-CELL ROWS ARE INSTANTLY FORCED
The most productive move in Number Blocks: find any row or column with exactly one empty cell. That cell's value is forced — it is simply the target sum minus the sum of the already-placed digits.
This is the Number Blocks equivalent of a naked single in Sudoku — and it fires far more often, because you're working with sums rather than unique-digit constraints. After placing a digit, check every row and column that contains that cell: any that now has one empty cell is instantly resolved.
Example: a row with target 18, containing placed digits 5 and 9 and one empty cell. The empty cell must be 18 − 5 − 9 = 4.
PROPAGATE BOTH AXES
Every placed digit affects two constraints: its row and its column. After placing a digit, don't just check the row — check the column too. A column might now have one empty cell and become instantly resolved. That resolution adds a digit to another row. That row might now be one-empty. Follow the cascade.
This bidirectional propagation is faster in Number Blocks than in most other sum puzzles because there are no combination constraints (any digit 1–9 is valid) — you're purely subtracting from the remaining targets and watching for the one-empty case.
RANGE NARROWING FOR MULTI-EMPTY ROWS
When a row still has two or more empty cells, you can't force a single digit — but you can narrow the range. Given the remaining target and the number of empty cells, compute the minimum and maximum possible sum for each cell:
- Minimum: if all other empty cells take their maximum value (9), what is the minimum this cell can be?
- Maximum: if all other empty cells take their minimum value (1), what is the maximum this cell can be?
If min = max, the digit is forced. If the range narrows to two or three values, check the column constraint to eliminate further. Range narrowing is the bridge between the easy one-empty placements and harder mid-game positions.
HOW IT COMPARES TO SUMPLETE AND KAKURO
All three — Number Blocks, Sumplete, and Kakuro — use row and column sum targets. The differences are significant:
- Sumplete starts with a pre-filled grid and asks you to delete digits. Number Blocks starts empty and asks you to fill.
- Kakuro requires unique digits (1–9) within each run. Number Blocks allows repetition — which makes the combination space larger but the one-empty-cell technique much faster.
Number Blocks is often the most beginner-friendly of the three because forced placements appear immediately and the cascade is visible without tracking digit combinations.
THE SOLVE ORDER
- Scan every row and column. Any with one empty cell: fill it immediately (target minus placed sum).
- After each placement, re-scan the affected row and column for new one-empty cases.
- When no one-empty rows remain, apply range narrowing to multi-empty rows and columns. Look for any cell where min = max.
- Use cross-axis constraints: if a cell's column range narrows its candidates, remove impossible values from its row's remaining target budget.
- Repeat until the grid is full.
Most beginner and intermediate Number Blocks grids fully resolve through one-empty scanning alone. Larger grids require range narrowing, but the core rhythm is the same: scan, place, propagate, repeat.